Preheat oven to 400F. Arrange a cooling or wire rack over a baking sheet.
Peel potatoes lengthwise, leave skin attached on ends of the potatoes. Cut each potato lengthwise into very thin fries no more than 1/4-inch thick the thinner the fries, the crispier the end product will be.
In a bowl, dress the fries lightly with EVOO, season with salt and pepper, and arrange in single layer on a baking sheet without crowding. Place fries in oven and roast 30 minutes. Remove fries and increase heat to 450F. Bring oven to new temp for about 10 minutes, then return fries to oven and bake another 10-15 minutes until very crisp and brown.
Meanwhile, melt butter in skillet over medium heat, add onions and garlic, thyme, salt and pepper, and cook 30 minutes, stirring occasionally. Be careful not to brown onions too fast you want them to be very soft and caramel in color. Deglaze with sherry then stir in stock to keep onions wet. Reduce heat to low and keep warm.
Combine ingredients for horseradish sauce in a bowl and reserve.
Season meat with salt and pepper and form 12 slider-size burgers, making them thinner at center than edges for even cooking.
Heat cast-iron pan or griddle to medium-high heat. When pan is very hot, drizzle burgers with a bit of oil and cook 5-6 minutes total for medium rare, a few minutes longer for medium well, turning occasionally. Melt cheese over burgers last minute of cooking time.
Place burgers on bun bottoms and top with onions, fries and horseradish sauce. Set bun tops in place and serve with spinach or green salad with Dijon vinaigrette alongside.